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  1. ❆r❦❤❛♥❣❡❧✬s❦✐ ✟ ✙ ❛❧♣❤❛ ♣r♦♣❡rt✐❡s ♦❢ C p ( X ) ❛♥❞ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ X ▲❡✈ ❇✉❦♦✈s❦ý ❚♦♣♦❧♦❣✐❝❛❧ ❙②♠♣♦s✐✉♠✱ Pr❛❣✉❡✱ ✷✵✶✻ ❏✉❧② ✶✶✱ ✷✵✶✻

  2. ❚❤❡ s❡q✉❡♥❝❡ s❡❧❡❝t✐♦♥ ♣r♦♣❡rt② ❙❙P ✱ s❡❡ ❬❙❝✷❪✱ s❛②s✿ ❢♦r ❛♥② ❛♥❞ ❢♦r ❛♥② s❡q✉❡♥❝❡ ♦❢ s❡q✉❡♥❝❡s ❝♦♥✈❡r❣✐♥❣ t♦ ✱ t❤❡r❡ ❡①✐sts ❛ s❡q✉❡♥❝❡ s✉❝❤ t❤❛t ✳ ❆ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ♣♦ss❡ss❡s t❤❡ ♣r♦♣❡rt② ✭ α i ✮ ✱ i = 1 , 2 ✱ s❡❡ ❬❆r❤❪✱ ✐❢ ❢♦r ❛♥② x ∈ X ❛♥❞ ❢♦r ❛♥② s❡q✉❡♥❝❡ {{ x n,m } ∞ m =0 } ∞ n =0 ♦❢ s❡q✉❡♥❝❡s ❝♦♥✈❡r❣✐♥❣ t♦ x ✱ t❤❡r❡ ❡①✐sts ❛ s❡q✉❡♥❝❡ { y m } ∞ m =0 s✉❝❤ t❤❛t lim m →∞ y m = x ❛♥❞ ✭ α 1 ✮ { x n,m : m ∈ ω } ⊆ ∗ { y m : m ∈ ω } ❢♦r ❡❛❝❤ n ✱ ✭ α 2 ✮ { x n,m : m ∈ ω } ∩ { y m : m ∈ ω } ✐s ✐♥✜♥✐t❡ ❢♦r ❡❛❝❤ n ✳ ■t ✐s ❦♥♦✇ t❤❛t ❢♦r ❈ p ( X ) t❤❡ ♣r♦♣❡rt✐❡s ✭ α 2 ✮ ✱ ✭ α 3 ✮ ❛♥❞ ✭ α 4 ✮ ❛r❡ ❡q✉✐✈❛❧❡♥t✱ s❡❡ ❬❙❝✸❪✳ ▲❡✈ ❇✉❦♦✈s❦ý ❚♦♣♦❧♦❣✐❝❛❧ ❙②♠♣♦s✐✉♠✱ Pr❛❣✉❡✱ ✷✵✶✻ ❆r❦❤❛♥❣❡❧✬s❦✐ ✟ ✙ ❛❧♣❤❛ ♣r♦♣❡rt✐❡s ♦❢ C p ( X ) ❛♥❞ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ X

  3. ❆ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ♣♦ss❡ss❡s t❤❡ ♣r♦♣❡rt② ✭ α i ✮ ✱ i = 1 , 2 ✱ s❡❡ ❬❆r❤❪✱ ✐❢ ❢♦r ❛♥② x ∈ X ❛♥❞ ❢♦r ❛♥② s❡q✉❡♥❝❡ {{ x n,m } ∞ m =0 } ∞ n =0 ♦❢ s❡q✉❡♥❝❡s ❝♦♥✈❡r❣✐♥❣ t♦ x ✱ t❤❡r❡ ❡①✐sts ❛ s❡q✉❡♥❝❡ { y m } ∞ m =0 s✉❝❤ t❤❛t lim m →∞ y m = x ❛♥❞ ✭ α 1 ✮ { x n,m : m ∈ ω } ⊆ ∗ { y m : m ∈ ω } ❢♦r ❡❛❝❤ n ✱ ✭ α 2 ✮ { x n,m : m ∈ ω } ∩ { y m : m ∈ ω } ✐s ✐♥✜♥✐t❡ ❢♦r ❡❛❝❤ n ✳ ■t ✐s ❦♥♦✇ t❤❛t ❢♦r ❈ p ( X ) t❤❡ ♣r♦♣❡rt✐❡s ✭ α 2 ✮ ✱ ✭ α 3 ✮ ❛♥❞ ✭ α 4 ✮ ❛r❡ ❡q✉✐✈❛❧❡♥t✱ s❡❡ ❬❙❝✸❪✳ ❚❤❡ s❡q✉❡♥❝❡ s❡❧❡❝t✐♦♥ ♣r♦♣❡rt② ❙❙P ✱ s❡❡ ❬❙❝✷❪✱ s❛②s✿ ❢♦r ❛♥② x ∈ X ❛♥❞ ❢♦r ❛♥② s❡q✉❡♥❝❡ {{ x n,m } ∞ n =0 ♦❢ s❡q✉❡♥❝❡s m =0 } ∞ ❝♦♥✈❡r❣✐♥❣ t♦ x ✱ t❤❡r❡ ❡①✐sts ❛ s❡q✉❡♥❝❡ { m n } ∞ n =0 s✉❝❤ t❤❛t lim n →∞ x n,m n = x ✳ ▲❡✈ ❇✉❦♦✈s❦ý ❚♦♣♦❧♦❣✐❝❛❧ ❙②♠♣♦s✐✉♠✱ Pr❛❣✉❡✱ ✷✵✶✻ ❆r❦❤❛♥❣❡❧✬s❦✐ ✟ ✙ ❛❧♣❤❛ ♣r♦♣❡rt✐❡s ♦❢ C p ( X ) ❛♥❞ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ X

  4. ❆ t♦♣♦❧♦❣♦❝❛❧ s♣❛❝❡ ✐s ❛ ✲s♣❛❝❡ ✐❢ ❋ ✳ ❚❤❡♦r❡♠ ✶ ✭❘❡❝➟❛✇ ❬❘❪✮ ❆♥② ♣❡r❢❡❝t❧② ♥♦r♠❛❧ ◗◆ ✲s♣❛❝❡ ✐s ❛ ✲s♣❛❝❡✳ ❈♦r♦❧❧❛r② ✷ ✭❇✉❦♦✈s❦ý ✕ ❘❡❝➟❛✇ ✕ ❘❡♣✐❝❦ý ❬❇❘❘❪✮ ❊✈❡r② s✉❜s❡t ✭✇✐t❤ t❤❡ s✉❜s❡t t♦♣♦❧♦❣②✮ ♦❢ ❛ ◗◆ ✲s♣❛❝❡ ✐s ❛ ◗◆ ✲s♣❛❝❡✳ ❆ s❡q✉❡♥❝❡ { f n } ∞ n =0 ♦❢ r❡❛❧ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ ❛ s❡t X q✉❛s✐✲♥♦r♠❛❧❧② ❝♦♥✈❡r❣❡s t♦ ❛ ❢✉♥❝t✐♦♥ f ✱ ✐❢ t❤❡r❡ ❡①✐sts ❛ s❡q✉❡♥❝❡ { ε n } ∞ n =0 ♦❢ ♥♦♥✲♥❡❣❛t✐✈❡ r❡❛❧s ❝♦♥✈❡r❣✐♥❣ t♦ 0 s✉❝❤ t❤❛t ( ∀ x ∈ X )( ∃ n 0 )( ∀ n ≥ n 0 ) | f n ( x ) − f ( x ) | ≤ ε n . ❆ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ✐s ❛ ◗◆✲s♣❛❝❡ ✐❢ ❡✈❡r② s❡q✉❡♥❝❡ { f n } ∞ n =0 ♦❢ ❝♦♥t✐♥✉♦✉s r❡❛❧ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ X ❝♦♥✈❡r❣✐♥❣ ♣♦✐♥t✇✐s❡ t♦ 0 ❛❧s♦ q✉❛s✐✲♥♦r♠❛❧❧② ❝♦♥✈❡r❣❡s t♦ 0 ✱ ❬❇❘❘❪✳ ▲❡✈ ❇✉❦♦✈s❦ý ❚♦♣♦❧♦❣✐❝❛❧ ❙②♠♣♦s✐✉♠✱ Pr❛❣✉❡✱ ✷✵✶✻ ❆r❦❤❛♥❣❡❧✬s❦✐ ✟ ✙ ❛❧♣❤❛ ♣r♦♣❡rt✐❡s ♦❢ C p ( X ) ❛♥❞ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ X

  5. ❆ s❡q✉❡♥❝❡ { f n } ∞ n =0 ♦❢ r❡❛❧ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ ❛ s❡t X q✉❛s✐✲♥♦r♠❛❧❧② ❝♦♥✈❡r❣❡s t♦ ❛ ❢✉♥❝t✐♦♥ f ✱ ✐❢ t❤❡r❡ ❡①✐sts ❛ s❡q✉❡♥❝❡ { ε n } ∞ n =0 ♦❢ ♥♦♥✲♥❡❣❛t✐✈❡ r❡❛❧s ❝♦♥✈❡r❣✐♥❣ t♦ 0 s✉❝❤ t❤❛t ( ∀ x ∈ X )( ∃ n 0 )( ∀ n ≥ n 0 ) | f n ( x ) − f ( x ) | ≤ ε n . ❆ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ✐s ❛ ◗◆✲s♣❛❝❡ ✐❢ ❡✈❡r② s❡q✉❡♥❝❡ { f n } ∞ n =0 ♦❢ ❝♦♥t✐♥✉♦✉s r❡❛❧ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ X ❝♦♥✈❡r❣✐♥❣ ♣♦✐♥t✇✐s❡ t♦ 0 ❛❧s♦ q✉❛s✐✲♥♦r♠❛❧❧② ❝♦♥✈❡r❣❡s t♦ 0 ✱ ❬❇❘❘❪✳ ❆ t♦♣♦❧♦❣♦❝❛❧ s♣❛❝❡ X ✐s ❛ σ ✲s♣❛❝❡ ✐❢ ❋ σ ( X ) = G δ ( X ) ✳ ❚❤❡♦r❡♠ ✶ ✭❘❡❝➟❛✇ ❬❘❪✮ ❆♥② ♣❡r❢❡❝t❧② ♥♦r♠❛❧ ◗◆ ✲s♣❛❝❡ ✐s ❛ σ ✲s♣❛❝❡✳ ❈♦r♦❧❧❛r② ✷ ✭❇✉❦♦✈s❦ý ✕ ❘❡❝➟❛✇ ✕ ❘❡♣✐❝❦ý ❬❇❘❘❪✮ ❊✈❡r② s✉❜s❡t ✭✇✐t❤ t❤❡ s✉❜s❡t t♦♣♦❧♦❣②✮ ♦❢ ❛ ◗◆ ✲s♣❛❝❡ ✐s ❛ ◗◆ ✲s♣❛❝❡✳ ▲❡✈ ❇✉❦♦✈s❦ý ❚♦♣♦❧♦❣✐❝❛❧ ❙②♠♣♦s✐✉♠✱ Pr❛❣✉❡✱ ✷✵✶✻ ❆r❦❤❛♥❣❡❧✬s❦✐ ✟ ✙ ❛❧♣❤❛ ♣r♦♣❡rt✐❡s ♦❢ C p ( X ) ❛♥❞ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ X

  6. ❆ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ ✐s ♣❡r❢❡❝t❧② ♠❡❛❣❡r ✐❢ ❡✈❡r② ♣❡r❢❡❝t s✉❜s❡t ♦❢ ✐s ♠❡❛❣❡r✳ ❚❤❡♦r❡♠ ✸ ✭❇✉❦♦✈s❦ý ✕ ❘❡❝➟❛✇ ✕ ❘❡♣✐❝❦ý ❬❇❘❘❪✮ ❆♥② ✇◗◆ ✲s♣❛❝❡ ✐s ♣❡r❢❡❝t❧② ♠❡❛❣❡r✳ ■❢ ✐s ♦♥❡ ♦❢ t❤❡ ♥♦t✐♦♥s ✭ ✮ ✱ ✱ ❙❙P✱ ✇◗◆✱ ◗◆✱ t❤❡♥ t❤❡ ♥♦t✐♦♥ ♦r ✐s ♦❜t❛✐♥❡❞ ❜② r❡♣❧❛❝✐♥❣ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ❜② ❧♦✇❡r ♦r ✉♣♣❡r s❡♠✐❝♦♥t✐♥✉♦✉s ♦♥❡s✱ r❡s♣❡❝t✐✈❡❧②✳ ❙❡❡ ❬❇❪✳ ■❢ ✐s ♦♥❡ ♦❢ t❤❡ ♥♦t✐♦♥s ✭ ✮ ✱ ✱ ❙❙P✱ t❤❡♥ t❤❡ ♥♦t✐♦♥ ✐s ♦❜t❛✐♥❡❞ ❜② r❡♣❧❛❝✐♥❣ ❛❧❧ ❝❛s❡s ♦❢ ♣♦✐♥t✇✐s❡ ❝♦♥✈❡r❣❡♥❝❡ ❜② q✉❛s✐✲♥♦r♠❛❧ ❝♦♥✈❡r❣❡♥❝❡✳ ❈♦♠♣❛r❡ ❬❇❙❪✳ ❆ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X ✐s ❛ ✇◗◆✲s♣❛❝❡ ✐❢ ❡✈❡r② s❡q✉❡♥❝❡ { f n } ∞ n =0 ♦❢ ❝♦♥t✐♥✉♦✉s r❡❛❧ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ X ❝♦♥✈❡r❣✐♥❣ ♣♦✐♥t✇✐s❡ t♦ 0 ❤❛s ❛ s✉❜s❡q✉❡♥❝❡ { f n k } ∞ k =0 q✉❛s✐✲♥♦r♠❛❧❧② ❝♦♥✈❡r❣✐♥❣ t♦ 0 ✱ ❬❇❘❘❪✳ ▲❡✈ ❇✉❦♦✈s❦ý ❚♦♣♦❧♦❣✐❝❛❧ ❙②♠♣♦s✐✉♠✱ Pr❛❣✉❡✱ ✷✵✶✻ ❆r❦❤❛♥❣❡❧✬s❦✐ ✟ ✙ ❛❧♣❤❛ ♣r♦♣❡rt✐❡s ♦❢ C p ( X ) ❛♥❞ ❝♦✈❡r✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ X

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